Existence and global attractivity of positive periodic solutions of functional differential equations with impulses

Abstract

Sufficient and realistic conditions are obtained for the existence and global attractivity of positive periodic solutions of the delay differential system with impulses d y ( t ) d t = y ( t ) F ( t , y ( t - τ 1 ( t ) ) , … , y ( t - τ n ( t ) ) ) , a.e. t > 0 , t ≠ t k , y ( t k + ) - y ( t k ) = b k y ( t k ) , k = 1 , 2 , … . The method involves the application of the Gaines and Mawhin's coincidence degree theory, the constructing suitable Lyapunov Functionals and estimations of uniform upper bounds on solutions. When these results are applied to some special delay population models, some new results are obtained, and some known results are generalized. In particular, our results indicate that under the appropriate linear periodic impulsive perturbations, the above impulsive delay differential equation preserves the original periodicity and global attractivity of the nonimpulsive delay differential equation d y ( t ) d t = y ( t ) F ( t , y ( t - τ 1 ( t ) ) , … , y ( t - τ n ( t ) ) ) .